Martin White1, Ludovic van Waerbeke2,3, Jonathan Mackey4 1Departments of Physics and Astronomy, University of California, Berkeley, CA 94720
2Institut d’Astrophysique de Paris, 98bis boulevard Arago, 75014 Paris France
3Canadian Institut for Theoretical Astrophysics, 60 St Georges St, Toronto, M5S 3H8 Ontario, Canada 4Harvard-Smithsonian CfA, 60 Garden St, Cambridge, MA 02138
email: [email protected] Draft version November 28, 2001
ABSTRACT
Using mock observations of numerical simulations, we investigate the completeness and efficiency of searches for galaxy clusters in weak lensing surveys. While it is possible to search for high mass objects directly as density enhancements using weak lensing, we find that line-of-sight projection effects can be quite serious. For the search methods that we consider, to obtain high completeness requires acceptance of a very high false-positive rate. The false positive rate can be reduced only by significantly degrading the completeness. Both completeness and efficiency are dependent upon the filter used in the search and the desired mass threshold, emphasizing that a measurement of the 3D mass function from gravitational lensing is affected by a number of biases which mix ‘cosmological’ and observational issues.
Subject headings: cosmology: theory – large-scale structure of Universe
1. INTRODUCTION
Clusters of galaxies are one of our most important cos-mological probes. As the most recent objects to form in the universe their number density and properties are exquisitely sensitive to our modeling assumptions. Their composition accurately reflects the mix of matter in the universe. They are bright and can be “easily” seen to large distances, allowing constraints on the crucial interval 0< z∼<1 where the universal expansion changes from de-celeration to acde-celeration. They are located close to their formation site. Being bright and sparse they are excel-lent tracers of the large-scale structure – they are highly biased so their clustering is easy to measure and is much more straightforwardly computed from theory than that of galaxies.
However, constructing large samples of galaxy clusters for statistical analyses remains a difficult task. For many years the only available catalogues were based on pro-jected galaxy overdensity, though it was quickly realized that such samples suffer from projection effects and the large scatter between optical richness and cluster mass (for recent theoretical studies see e.g. van Haarlem, Frenk & White 1997; Reblinsky & Bartelmann 1999 White & Kochanek 2001). At present cluster samples have been constructed using optical, X-ray and Sunyaev-Zel’dovich ‘surveys’ each with their own advantages and disadvan-tages.
In the last few years it has become possible to search for high mass objects directly as density enhancements using weak gravitational lensing. Since the first detections by Bonnet, Mellier & Fort (1994) and Fahlman et al. (1994), this technique has been demonstrated many times on pre-viously known clusters (see Wu et al. 1998 and Hattori et al. 1999 for recent reviews). Erben et al. (2000) and Umetsu & Futamase (2000) have reported ‘dark clumps’, where mass concentrations are seen with no optical or X-ray detection of a cluster. To date there is only one case of a confirmed cluster discovered first with weak gravitational
lensing: Wittman et al. (2001) serendipitously detected a z= 0.276 cluster withσv∼600km/s in the corner of one of the fields they used previously for cosmic shear.
Thus lensing offers a completely new way to find clus-ters. As with all new methods it avoids some of the prob-lems which plague other methods but has its own system-atics. In this paper we want to look at the power and pitfalls of this method. Specifically we want to look at how efficient and complete weak lensing surveys are at constructing a mass selected sample of clusters.
2. SIMULATED OBSERVATIONS
2.1. N-body simulation
We wish to make simulated convergence maps which are as realistic as possible while at the same time knowing the real 3D location of any clusters in the field. Our pro-gramme begins with a model for the evolution of the dark matter which governs the formation of large-scale struc-ture. On Mpc scales we expect that the baryonic matter will faithfully trace the dark matter, thus our model should reproduce the spatial distribution of mass. This problem can be well tackled by modern numerical simulations. We have run a 2563 particle simulation of a ΛCDM model in a 200h−1Mpc box using the TreePM-SPH code (White et al. 2001) operating in collisionless (dark matter only) mode. This simulation represents a large cosmological vol-ume, to include a fair sample of rich clusters, while main-taining enough mass resolution to identify galactic mass halos. Because it provides a reasonable fit to a wide range of observations, including the present day abundance of rich clusters of galaxies (Pierpaoli, Scott & White 2001), we have simulated the “concordance cosmology”, which has Ωm = 0.3, ΩΛ = 0.7, H0 = 100hkms−1Mpc−1 with h= 0.67, ΩB= 0.04,n= 1 andσ8= 0.9 (corresponding to δH = 5.02×10−5). The simulation was started atz= 50 and evolved to the present with the full phase space dis-tribution dumped every 100h−1Mpc fromz'2 toz= 0. The gravitational force softening was of a spline form, with 1
Fig. 1.—(top left) A simulatedκmap, 3◦on a side, with a linear greyscale which maps all pixels withκ <0 to white. (top right) The same field with noise added. (bottom left) The noisy map, smoothed with a Gaussian of about 10. (bottom right) TheMap map obtained from the noisy map with a filter of about 40.
a “Plummer-equivalent” softening length of 28h−1kpc co-moving. The particle mass is 4×1010h−1M allowing us
to find bound halos with masses several times 1011h−1M
and giving many, many particles in a cluster mass halo (>1014h−1M) to begin to resolve substructure.
For every output of the simulation we produce a halo catalogue by running a “friends-of-friends” group finder with a linking length b = 0.15. We use this rather than the more canonicalb= 0.2 as we find that the larger link-ing length frequently merges what we would by eye list as separate halos. By effectively ‘removing’ one of the halos from our list this would have a deleterious impact upon our efficiency calculations. While this effect is not entirely eliminated when usingb= 0.15, it is significantly reduced. We keep all groups above 64 particles, which imposes a minimum halo mass of order 1012h−1M. For each
iden-tified halo we compute, using the 3D distribution of all of the particles in the simulation, the mass (we useM200, the mass enclosed within a radius,r200, within which the mean density is 200 times thecritical density at that red-shift), velocity dispersion etc. so we can understand our selection in terms of the intrinsic, rather than projected, cluster properties.
2.2. The convergence map
We simulate an observed field by “stacking” different slices through the box at earlier and earlier output times. We divide every output up into 6 halves (top, bottom, left, right, front, back) of 200×200×100h−1Mpc. A given observational field is then obtained by choosing, ev-ery 100h−1Mpc along the line-of-sight, a random half of the box at that output, shifted perpendicular to the line-of-sight by a random amount using the periodicity of the simulation volume. All of the mass in that half of the box is projected onto the sky. The convergence is
κ= 3
2Ωmat(H0D∗) 2Z
dt w(t) δ
a (1)
Fig. 2.—The ‘projected’ mass function of clusters in our five 3◦×3◦simulated lensing fields. The squares indicate all clusters lying between the source and the observer in the fields, the triangles those for which the lensing kernel,t(1−t), is greater than half its peak value.
whereD∗ is the (comoving) angular diameter distance,
dD da ≡
1
a2H(a) , (2)
to a fiducial point beyond the furthest source, t=D/D∗
is a dimensionless distance,w(t) is a weight function w(t) = Z 1 t dts dn dts t(ts−t) ts (3)
which reduces tot(1−t) if all of the sources are at a fixed distance D∗ andδ =ρ/ρ¯−1. We approximate this
inte-gral as a weighted sum of the projected mass Σ in each plane, with the projection being done parallel to the edges of the simulation volume. Studies by Hamana, Colombi & Mellier (2000) have indicated that this method produces results which are almost identical to more numerically in-tensive ray tracing simulations. We have found that there is a slight positive bias in the power spectrum ofκusing this method as compared to the “tiling” method of White & Hu (2000) which does not approximate the path with segments parallel to the box boundaries. This small bias will not affect any of our conclusions.
We have chosen 100h−1Mpc as our sampling interval be-cause it is large enough that edge effects are minimal even for rich clusters while being fine enough that line-of-sight integrals are well approximated by sums over the (static) outputs. However, even though only a small fraction of clusters lie within r200 of a slice boundary, we decided to require that the orientation and offset change only on ev-ery second slice. Thus if we choose at one redshift the front of the box the next slice is required to be the back. In this manner a cluster on the boundary is almost al-ways included, but the structure still evolves in steps of 100h−1Mpc not 200h−1Mpc.
As a first step we have generated 5 maps, each 3◦×3◦, with sources fixed at zs = 1 (one example is shown in Fig. 1). These maps are only approximately independent as they are drawn from the same simulation, but the ran-dom orientations allows us to sample different possible pro-jection effects. For each map we can add Gaussian pixel noise at a specified level and we smooth the maps (us-ing fourier transform methods) with a Gaussian beam of a specified FWHM or convolve the map with some other filter.
The same random slices and offsets are used to project the group catalogue into the field, and to produce 3D im-ages1 of the field of view from the particle distribution (which we use to check for projection effects). We shall denote by ‘cluster’ any halo having M200 > 1014h−1M
and our catalog includes all such halos lying in the field with z ≤ 1. We define the center of a cluster as the po-sition of the potential minimum, calculating the potential using only the particles in the FoF group. This proved to be more robust than using the center of mass, as the poten-tial minimum coincided closely with the density maximum for all but the most disturbed clusters. The mass function of clusters lying between the source and observer in our 5 simulated fields is shown in Fig. 2. The distribution of angular sizes is shown in Fig. 3.
1The particle distribution is visualized usingTipsy from the N-body shop at the University of Washington.
2.3. Finding & matching peaks
For a given (possibly noisy and smoothed) map we need an algorithm for finding peaks. We have chosen a simple strategy whereby we search the map for all pixels which are a local maximum. This set forms our base peak list, which we number. We then search around each maximum and include the adjacent pixels if κ > f κmax where f is a user specified fraction typically set to 0.7. The peaks are all extended at the same rate, so that adjacent peaks do not swallow each other. This algorithm then returns for every pixel in the map the peak number (or possibly “no peak”) to which it belongs. For each peak we keep track of both the maximum κand the sum, κsum, of the convergence in all pixels above the thresholdf κmax. The latter is loosely correlated with mass.
In addition to our ‘peak list’ we have our ‘cluster list’ from the 3D cluster catalogues. Given the very large num-ber of both peaks and clusters, and the lack of distance information in the peak list, matching these can be quite problematic. We perform the match in two directions: whether a peak is in the cluster list (forward match) and whether a cluster is in our peak list (backward match). We typically find that all clusters above 1014h−1M lie
within ±1 pixel of an extended peak, but some clusters lie near peaks (local maxima) with very low values of κ (see below). Because each of the lists is so long and we are only using 2D information in associating peaks to clus-ters, a ‘match’ is claimed only if the forward and backward associations agree.
The code produces a list of peaks and halos with matches flagged (plus cases with only a forward or backward match). There are two key numbers which we shall focus on below. The first is the fraction of peaks which matched at least one halo, which will determine our ‘efficiency’. The second is the fraction of halos which matched at least one peak, which will determine our ‘completeness’.
Note that as we begin to smooth the maps and add noise, the 1-1 correspondence between peaks and halos will begin to degrade. We take the attitude that all potential
Fig. 3.—The distribution of projected virial radii for the clusters in our five 3◦×3◦simulated lensing fields withM200>1014h−1M.
detections would be followed up with e.g. X-ray observa-tions or redshifts. Thus if two halos match a single ex-tended peak we can count those both as detections since followup of that area of sky would presumably find both of them.
Our method is very robust and can be easily automated, but it is not perfect. In particular it can cause us to under-estimate the efficiency of a lensing search due to the way we define our halos. For example, if two massive halos are close together (perhaps in the initial stages of a merger or interaction) they can be linked by our group finder into a single halo. Our group catalog will then contain param-eters only for the mass around the potential minimum, missing the other halo entirely. This halo still has a large amount of mass associated with it however, and is quite overdense, so it will likely correspond to a κ peak. This peak will be erroneously counted as a miss as there is no corresponding entry in the cluster list.
This effect is mitigated to a large extent by the rela-tively small linking length we have chosen to define our 3D catalog. Neighboring halos are linked only if the ma-terial between them is ∼>102×overdense. We have found one example of such an artificial linking for systems of cluster mass, corresponding to a chain of halos lying along a filament. This single system changes our results by less than a percent. Inspection of other ‘strange’ peaks has not yielded any other examples of this effect.
3. RESULTS
3.1. Raw maps
We first present results on the raw κ maps, i.e. with-out adding noise or smoothing the map. These results will set a theoretical ‘best’ performance and allow us to under-stand how much things are degraded by noise and smooth-ing. Our results for the noise-free, unsmoothed maps are listed in Table 1. We see that there are a very large num-ber of peaks, and that almost all clusters can be matched to peaks (we discuss the rare misses below). If we apply a
Fig. 4.—The ratio of the “lensing mass” to the true mass for clusters above 1014h−1Mfor 5 of our 3◦×3◦lensing fields. The solid histogram is all clusters and the dashed line is those for which
Fig. 5.—A zoom in of one of our convergence maps showing two clusters which lie almost on top of each other in projection. (left) the fullκmap. (middle) the portion coming from the 100h−1Mpc slice at low redshift. (right) the portion coming from the 100h−1Mpc slice at higher redshift.
threshold to our peak finder to reduce the peaks to a rea-sonable number our completeness begins to drop rapidly.
The reason for this can be seen by considering the dis-tribution ofκvalues in the region of known clusters. We have chosen to do this in the context of cluster mass mea-surements, realizing that this issue has been discussed in many contexts previously. As a first step we went through our cluster list and for all clusters above 1014h−1M we
summed the values of κ within a disk centered on the known cluster center and with radius equal to the known value ofr200. Using the cluster redshift we then converted this to a mass. A histogram of the mass compared to the true mass is shown in Fig. 4 where we see that there is both a large scatter, as has been emphasized before2, and a positive offset. Perhaps the most dramatic are the few clusters with a negative mass. These arise when a cluster in a region where the lensing kernel is small has a large void projected along the line-of-sight at a distance where the lensing kernel is large. This leads to a negative meanκ even though there exists a significant mass overdensity. To isolate this effect we further require that the lensing kernel at the cluster position be more than half of its peak value. This largely removes the negative tail, although even with
2See for example the discussion in§7.3 of Metzler, White & Loken (1999). Peaks Clusters w >0 w > wmax/2 All 96418 0.99 0.99 >0.0 72295 0.99 0.99 >0.2 603 0.52 0.65 >0.4 37 0.07 0.09 Table 1
Peak statistics for the ‘raw’ maps, as a function of thresholdκmax. The columns give the number of peaks above the thresholds listed, and the fraction of clusters
(halos with M200>1014h−1M) satisfying the lensing kernel cut which were found by matching to those peaks. There were 540 and 419 clusters withw >0andw > wmax/2
respectively.
this cut 4 clusters (out of more than 400) with Mκ < 0 remain.
The tail to very large mass ratios occurs when a massive cluster is projected on top of a less massive cluster. These lines-of-sight were excluded in the analysis of Metzler et al. (1999), but we have not done so here. Thus our pro-cedure will return something close to the cumulative mass for each of these systems, which will be a large overesti-mate for the low mass system. We show an example of such an overlap in Fig. 5. Again, restricting ourselves to clusters near the peak of the lensing kernel reduces this ef-fect. We shall return to the projection effects on the mass measurement of the detected peaks at the end of§3.3.
While the details change, this kind of effect is also seen in the distribution of peak κvalues or in other measures of ‘mass’ based on lensing.
3.2. Adding noise
While the above results can serve to indicate some of the pitfalls in weak lensing searches for clusters, they do not indicate how a realistic search would perform. To make further steps in this direction we need to include ‘noise’ in our maps. Following van Waerbeke (2000) we model the noise in aκmap that would arise from processing a shear map using a technique such as that of Kaiser & Squires (1993) as Gaussian, correlated only by any smoothing ker-nel applied to the map. If the mean intrinsic ellipticity of the source galaxies isγintthen the noise introduced in our κmap has variance
σ2pix= γ 2 int ¯ nθ2pix (4) where ¯nis the mean number density of sources. Assuming γint ∼0.2 and ¯n '105deg−2 we haveσpix '0.1 for our 3◦/512 pixels. This level of noise is quite optimistic, com-pared to current observations, but none of our conclusions will depend on this specific choice, though the absolute meaning of our signal-to-noise cuts will clearly scale with σpix.
Even this level of noise is quite large compared to our signal, so we need to smooth the maps to enhance the
contrast of our signal to the noise. We have not at-tempted to search for an ‘optimal’ filter, matched to the predicted shape of the cluster. Instead we have chosen to either smooth the maps with a Gaussian whose FWHM is roughly matched to the size of clusters at cosmological dis-tances, 10−20, or to apply theMapfilter already discussed in the literature (see§3.3).
Fig. 6 shows how a survey’s completeness and efficiency depend on the smoothing scale in the case of no noise. As expected increasing the smoothing decreases the number of false positives and thus the expense of follow up observa-tions; but it also decreases the survey completeness. The best match of peak threshold and smoothing will depend on the trade offs between these two issues. The trade off is also affected by the level of noise in the map, as is shown in Fig. 7. Note that for no combination of parameters is it possible to have>50% completeness with<50% contam-ination! This may be traced primarily to the large scatter in the mass-peak relation shown in Fig. 4 (for a discussion of precisely this effect in redshift surveys for clusters, see White & Kochanek 2001).
We note that only for very high thresholds is our effi-ciency is not 100%. This means that there are quite promi-nent peaks in the maps which do not match any cluster with M200 > 1014h−1M. This could have relevance to the question of ‘dark clusters’ (see Fischer 1999, Erben et al. 2000, Umetsu & Futamase 2000).
3.3. Matched filter
One can enhance the signal-to-noise for cluster-like struc-tures by convolving theκmap with a ‘matched filter’. For a certain class of such filters, known as aperture mass mea-sures (Schneider 1996; Schneider et al. 1998) this operation is easy to implement directly on the shear field itself: con-volution of theκ map with a kernel U can be shown to be the same as convolving the tangential shear map with
Fig. 6.—Completeness and efficiency as a function of smooth-ing for maps with no noise. Solid lines are the fraction of clus-ters (groups withM200 > 1014h−1M) in the 5 fields which are matched (completeness), dashed lines are the fraction of identified peaks which correspond to clusters (efficiency). The symbol types denote the κ threshold for counting peaks: (squares) κmax > 0, (triangles)κmax>0.1, (circles)κmax>0.2.
a related kernelQprovided the kernelU vanishes outside of some radiushand is compensated, viz
Z
θdθ U(θ) = 0 . (5) The convolution of κwith U to produce the Map map is a bandpass filter on the convergence map, with the filter function quite narrow in (spatial) frequency space (Bartel-mann & Schneider 1999). Thus we may expect that for an appropriately chosen Map scale, h, we will enhance the contrast of clusters in the map.
We have created, from our noisyκmaps, a series ofMap maps with different filtering scales. We use the simplest, `= 1,Map kernel θ2U(θ) = 9 π(1−u 2 ) 1 3 −u 2 if θ < h. (6) Here u ≡ θ/h and h is the filter size. We perform this convolution directly on the κ map, with Map filters for whichhis a multiple of the pixel scale. A typical cluster is ∼ 10 pixels across, so pixelization effects are not too severe on the scales of interest. Convolution of theκmap with this kernel indeed enhances the visibility of clusters in the maps as can be seen in Fig. 1. Finally we produce S−statistic maps by dividing ourMapmaps by their rms fluctuation.
Using these maps as the input to our peak finding soft-ware we find that the efficiency and completeness depend on the filter size in the expected manner. Recalling that theMapfilter scale is roughly 3×the Gaussian width for similarly extended kernels the results in Fig. 8 can be seen to be quite similar. In the presence of noise the ‘optimal’ filtering scale (30−40) is slightly larger than for the noise free map. One could argue that the low completeness levels we are finding are a result of our mass threshold, and that we would find a larger fraction of the higher mass clusters. This is in fact not true, Table 2 shows the completeness and efficiency for different cuts keeping only clusters above 3×1014h−1M. There is not a significant improvement
Fig. 7.—As for Fig. 6, but with an rms pixel noise of 0.1 added to theκmap prior to smoothing.
compared to the 1014h−1M mass threshold. Finally we
have also restricted ourselves to the clusters which lie in regions where the lensing kernel is more than half of its peak value. These clusters clearly have a larger proba-bility of contributing to a significant peak than clusters near the observer or the sources. The completeness and efficiency numbers are given in Table 3 and tell a similar story to the figures.
Fig. 8 shows that even quite highS−peaks can be ineffi-cient or incomplete. This does not mean that these peaks are purely noise however. In many cases a strong peak can be found to match to a lower mass object (or objects) along the line-of-sight. To illustrate this we have matched all groups above 1013h−1M with all peaks havingS >4
in the 5 fields. Fig. 9 shows a scatter-plot of all the unique matches. Notice that there is a substantial tail of lower mass groups even forS > 4. It is these low-mass groups that are driving our low efficiencies.
Our low efficiency is caused by ‘noise’ coming from both intrinsic ellipticities and from cosmic structures. The mea-sured mass function will therefore be biased toward the most massive objects. This problem could be overcome if we knew the mass Probability Distribution Function of those noise peaks, which we could try to deconvolve from the measured mass function. Unfortunately, only the noise peakS-distribution is known, as this can be derived an-alytically given the ellipticity dispersion (van Waerbeke 2000). It is interesting to ask whether this problem could be suppressed with an arbitrary low noise mass map, such as could be obtained by observing fields with a very long exposure time.
To answer this we measured the mass function associ-ated with all peaks in the noise-free fields which do not match any real cluster above 1014h−1M. In order to
as-sign a mass to a fake peak, we followed Erben et al. (2000) and computed theminimum mass that would naively be
Fig. 8.—Completeness and efficiency as a function of filtering scale forS−statistic maps with noise added as in Fig. 7. Solid lines are the fraction of clusters in the 5 fields which are matched (com-pleteness), dashed lines are the fraction of identified peaks which correspond to clusters (efficiency). The symbol types denote theS threshold for counting peaks: (squares)S >1, (triangles) S >3, (circles)S >5.
Scale S >1 S >3 S >5 (arcmin) Com Eff Com Eff Com Eff
1.1 98 0 85 4 45 34 2.1 98 0 85 4 45 34 3.2 96 1 79 6 51 39 4.2 98 1 75 9 49 50 5.3 89 1 68 12 40 53 Table 2
The completeness and efficiency (to the nearest per cent) for clusters aboveM200= 3×1014h−1Mas a function of
filter scale. There are only 47 such clusters in the 5 fields, so the statistics are considerably poorer than for
our fiducialM200= 1014h−1M.
Scale S >1 S >3 S >5 (arcmin) Com Eff Com Eff Com Eff
1.1 95 1 43 10 4 80 2.1 92 2 58 24 12 79 3.2 91 4 54 39 14 90 4.3 88 7 48 51 12 91 5.3 79 9 39 57 9 81 Table 3
The completeness and efficiency (to the nearest per cent) for clusters withw > wmax/2as a function of filter scale. There are 419 clusters in this restricted distance interval
in the 5 fields.
Fig. 9.—A scatter plot of unique matches of all groups with
M200 >1013h−1Mand all peaks withS >4 in our 5 maps with a filtering scale of'30. We have divided the line-of-sight into three intervals in distance. Open squares mark the closest 1/3, crosses the middle 1/3 and filled triangles the most distant 1/3 of the distance to the source. Note that there are a substantial number of objects below our chosen mass threshold (1014h−1M) for clusters.
associated with the peak. This mass corresponds to the deflector being at the redshift where the lensing kernel peaks (z'0.5 in our case) and is a conservative way for observers to give a lower mass to what they believe to be a real cluster. Fig. 10 shows this minimum mass func-tion of the missed peaks (empty symbols) compared to the true mass function (filled circles). The estimated mass depends of course on the aperture one uses. The three sets of empty symbols show the mass function obtained using three different aperture sizes: we used a square box of width 2.10, 2.80 and 3.50, corresponding respectively to 0.7, 1, and 1.2h−1Mpc radius at the assumed lens redshift of 0.5. This choice matches the typical aperture size used in the literature, and correspond to the virial radii used to compute most of the masses in this work (Fig. 3). We see that the non-matched peaks contribute significantly to the mass function below 3×1014h−1M. This means that even in the ideal case of noise free data, a large fraction of “clusters” in the range [1014h−1M, 3×1014h−1M] are only projections of large-scale structure. Lowering the mass threshold does not significantly change Fig. 10.
Phrased another way, the presence of a distribution of halos and large-scale structures provides a fluctuating background to ourκmaps. The clusters we are seeking are embedded in this background, which has a similar effect on the mass function (broadening it) as does measurement noise. However this broadening depends on the r.m.s. of the mass map, which depends on the cosmological model one considers. Thus in order to recover the cluster mass function by deconvolution we would need to assume an underlying cosmological model.
Fig. 10.—The mass functions of ‘real’ and ‘missed’ peaks. Filled circles show the mass function of 3D clusters in the simulation vol-ume. Empty symbols show the minimum estimated mass of the missed peaks (see text) which do not match a true cluster with mass above 1014h−1M. The mass of these peaks is computed in a square box of width 2.10(triangles), 2.80(squares) and 3.50 (dia-monds).
Also note that Fig. 10 suggests that it is possible to obtain peaks in a lensing map which can be interpreted at structures as massive as a few 1014h−1M due simply
to projection effects. This may bear upon the reports of ‘dark clumps’ with masses around a few 1014h−1M by (Fischer 1999, Erben et al. 2000, and Umetsu & Futamase 2000). In these papers, the authors computed the proba-bility for theSpeak to be ‘real’ against random alignment of lensed galaxies, but they neglected the effect of projec-tion of lower mass clumps discussed here.
We show in Fig. 11 that the efficiency and complete-ness are not equally distributed among the different clus-ter masses. Above a few 1014h−1M we are approach-ing 100% completeness whatever the chosen S threshold, i.e. we do find all sufficiently massive clusters. The strange drop in completeness at 4×1014h−1M is due to small
number statistics and reflects the limitations of our sim-ulation. Our original box has very few clusters above 4×1014h−1Mand an abnormally high percentage of them happened to lie at very high-z in the fields we simulated. They were therefore moved to smaller S than their mass might suggest due to the low lensing efficiency near the source, and they fall below our S cut. We have checked that this result does not depend on the smoothing kernel, we obtain the same behavior with the cluster mass whether we use 10 or 50 and it reflects thez-distribution of the few very high mass clusters.
We show the correlation between distance and S in Fig. 12. This correlation is easy to understand: at a fixed mass lowSclusters correspond preferentially to a low lens-ing efficiency, thus they are at either high or at low red-shift. Since the cosmological volume is much larger at high than at low redshift, we naturally catch more high redshift clusters. This also points out to a problem in using a fixed S threshold for cluster selection: it will affect the redshift selection function, and this should be properly taken into account in cluster abundance analysis for instance.
4. COMPARISON WITH PREVIOUS WORK
Ours is not the first study to investigate how complete a weak lensing survey of clusters could be using numerical simulations. It does however improve upon early work in some respects. The closest predecessor to our work, in terms of focus, is that of Reblinsky & Bartelmann (1999). These authors used N-body simulations to look at pro-jection effects in weak lensing selected cluster samples, and compared it to the projection effects in richness selected clusters. They generated a 3D cluster catalog from a sin-gle output of one of the GIF simulations in an 85h−1Mpc box. Their weak lensing maps were constructed by taking a projection of the mass in the box, constructing the 2D lensing potential to calculate the shear, and use aperture mass (Map) methods to find peaks. Since we start with a simulation volume 13 times larger (containing many more clusters) and simulate the entire ∼ 2000h−1Mpc line-of-sight, not just one 85h−1Mpc piece of it, we could study projection effects over cosmological scales. It turns out that this exacerbates the projection effects seen in their work and makes our results slightly more pessimistic than theirs.
Reblinsky & Bartelmann (1999) worked only at 20. Our efficiency/completeness at 20(Fig. 8) looks similar to their result, although it is difficult to compare the results
di-rectly since Reblinsky & Bartelmann (1999) give only cu-mulative numbers. Both sets of results can be explained by the leakage of small mass peaks to higher masses due to the convolution of the true cluster mass function with the ‘noise’. This predicts that we should observe more high mass clusters than there really are, in accord with our re-sults. Here we have demonstrated how this result depends on the smoothing kernel used, with the dependence being non-trivial. In particular we found that beyond a given scale (which is kernel dependent) increasing the smooth-ing scale always pushes the efficiency close to 100% (but the price to pay is a low completeness).
Simulations which include the entire line-of-sight have been performed by Reblinsky, Kruse, Jain & Schneider (1999), following up the semi-analytic work of Kruse & Schneider (1999). These authors made “Mapmaps” from simulated shear maps, with and without adding random noise. From these they extracted the number density of Mappeaks withS/N >5 (where for the map with no noise they estimated the noise due to intrinsic galaxy elliptici-ties). They find quite good agreement with the analytic estimates, in that∼5 peaks per square degree are found above this detection threshold for the map with no noise3. Unfortunately, due to the lack of knowledge of the 3D clus-ter positions in their analysis, it was impossible for them to study the completeness/efficiency. This makes it im-possible to compare directly with our work.
5. DISCUSSION
In the last few years it has become possible to search for clusters of galaxies directly as mass enhancements using weak gravitational lensing. This method probes the mass of a cluster independent of its dynamical state, and thus
3Note: this doesnot imply that the survey is complete, simply that the analytic estimate produces the same fraction of identified clusters as the simulation.
Fig. 11.— Efficiency and completeness as a function of mass for maps with a'30 filtering scale. The ‘dip’ at 4×1014h−1M is a statistical fluctuation due to the redshift distribution of the very small number of clusters withM200>4×1014h−1Min the simulated fields (see text). Solid lines show completeness, dashed lines efficiency. Solid squares are forS >3, trianglesS >4 and open circlesS >5.
presents a different view to surveys based on galaxy counts or the intra-cluster medium. For this reason a lens selected survey of clusters is an appealing sample, which could in principle be mass selected allowing a reconstruction of the cluster mass function with redshift.
Unfortunately there are many obstacles to be overcome. Our study strongly implies that complementary observa-tions (both weak and strong lensing, optical, Sunyaev-Zel’dovich, X-ray) will be of great help in cleaning a sample of lensing selected clusters of spurious detection and pro-jection effects (e.g. Castander 2000; Bartelmann 2001). An example of this is the confirmation using redshifts of the lensing selected cluster by Wittman et al. (2001). This will probably involve a significant number of followup observa-tions on lensing (mass) preselected clusters. In this paper we have begun to address the problem of designing such a survey, depending on the different goals (completeness, cluster redshift, mass range) one wants to achieve.
Extending upon the work of Reblinsky & Bartelmann (1999) and Metzler et al. (1999), we studied the problems associated with selecting clusters in lensing data, using larger numerical simulations of clusters which simulated the entire past light-cone. We focussed on the aperture mass statistic, assuming one identified source population, and investigated its dependence on cluster mass and red-shift. We compared the catalogs produced by different lensing-based analyses to the reference set of 3D clusters present in the simulation.
As discussed before, measurements of the masses of in-dividual clusters from weak lensing have a large scatter (100%) and a significant bias (about 20%), for clusters more massive than 1014h−1M (see Fig. 4). The bias
comes from the fact that clusters live preferentially in larger structures. The large scatter is due to the pres-ence of a large number of halos of different masses mak-ing up the large-scale structure. Phrased in terms of a mass, the ‘noise’ induced by this structure is comparable to the signal from a cluster of 1014h−1M put at redshift
Fig. 12.—A scatter-plot showing distance andSfor all clusters in our 5 fields with mass above 2×1014h−1Mand S >1. The more distant clusters, having a lower lensing efficiency, have lower
of∼0.5. This may bear upon the existence of dark clus-ters with mass of a few 1014h−1M in lensing surveys,
however to be sure whether projection effects are the ex-planation would require simulating the distribution of the light (i.e. galaxies) under the same observational condi-tions. If light suffers less projection than mass, this may explain the dark clusters. In agreement with Hoekstra (2001) we find that the noise due to uncorrelated large-scale structure along the line-of-sight does not obscure the signal from sufficiently massive (∼1015h−1M) clusters.
This ‘mass scatter’ biases the mass function measured with lensing. The scatter contains an intrinsic contribu-tion, which is cosmological model dependent, and a mea-surement noise contribution determined by the intrinsic galaxy ellipticities and the smoothing kernel. It is possi-ble to model the noise in the S statistic (and to predict the number density of observedS peaks as in Reblinsky et al. (1999), for instance), but it is much more difficult to transpose thisS-noise into a mass noise since S is not simply related to the mass (see Fig. 9), and this relation depends on the cosmological model.
Even if we restrict ourselves to the methods explored here, lensing surveys should be complete for the highest mass clusters, (∼> 3 ×1014h−1M) with reasonable
effi-ciency (0.1 −0.5 if S > 5 for example). However it is important to take into account the variation of the lensing kernel with redshift in interpreting the mass threshold of anS-selected sample.
It is not clear to what extent these issues can be over-come. In this work, we have not made use of filters matched to the cluster profiles or of multiple source popu-lations. In principle incorporating either of these could in-crease the efficiency or completeness of our samples. How-ever, at the very least these issues have to be considered in projects aimed at performing a statistical measure of clus-ter masses from weak lensing. In particular the trade-off between completeness/efficiency and mass bias are impor-tant aspects for plans to measure the lower mass clumps, like groups of galaxies (Schneider & Kneib 1998; M¨oller et al. 2001).
ACKNOWLEDGEMENTS
We would like to thank Peter Schneider for comments on an earlier draft. This work was supported in part by the Alfred P. Sloan Foundation and the National Science Foundation, through grants PHY-0096151, ACI96-19019 and AST-9803137.
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